Intrinsic geometry and curvature

In the previous chapters of this book we have frequently used the concept ‘Riemannian space’ or ‘curved space’. Except in Section 14.4 on the geodesic deviation, it has not yet played any rôle whether we were dealing only with a Minkowski space with complicated curvilinear coordinates or with a genuine curved space. We shall now turn to the question of how to obtain a measure for the deviation of the space from a Minkowski space.

If one uses the word ‘curvature’ for this deviation, one most often has in mind the picture of a two-dimensional surface in a three-dimensional space; that is, one judges the properties of a two-dimensional space (the surface) from the standpoint of a flat space of higher dimensionality. This way of looking at things is certainly possible mathematically for a four-dimensional Riemannian space as well – one could regard it as a hypersurface in a ten-dimensional flat space. But this higher-dimensional space has no physical meaning and is no more easy to grasp or comprehend than the four-dimensional Riemannian space. Rather, we shall describe the properties of our space-time by four-dimensional concepts alone – we shall study ‘intrinsic geometry’. In the picture of the two-dimensional surface we must therefore behave like two-dimensional beings, for whom the third dimension is inaccessible both practically and theoretically, and who can base assertions about the geometry of their surface through measurements on the surface alone.

The picture of black holes we have drawn so far changes drastically if quantum effects are taken into account. Before we go into the details of this in Section 5 of this chapter, we want to make a few general remarks on the interplay of Relativity Theory and Quantum Theory. For a more detailed discussion we refer the reader to the literature given at the end of the chapter.

The problem

The General Theory of Relativity is completely compatible with all other classical theories. Even if the details of the coupling of a classical field (Maxwell, Dirac, neutrino or Klein–Gordon field) to the metric field are not always free of arbitrariness and cannot yet be experimentally tested with sufficient accuracy, no doubt exists as to the inner consistency of the procedure.

This optimistic picture becomes somewhat clouded when one appreciates that besides the gravitational field the only observable classical field in our universe is the Maxwell field, while the many other interactions between the building blocks of matter can only be described with the aid of Quantum Theory. A unification of Relativity Theory and Quantum theory has not yet been achieved, however.

One of the main postulates of relativity theory is that a locally geodesic coordinate system can be introduced at every point of space-time, so that the action of the gravitational force becomes locally ineffective and the space is approximately a Minkowski space. Hence it is easily understandable why in our neighbourhood, with its relatively small space curvature, space is, to very good approximation, as it is assumed to be in quantum theory.

The problem

When we are handling physical problems, symmetric systems have not only the advantage of a certain simplicity, or even beauty, but also special physical effects frequently occur then. One can therefore expect in General Relativity, too, that when a high degree of symmetry is present the field equations are easier to solve and that the resulting solutions possess special properties.

Our first problem is to define what we mean by a symmetry of a Riemannian space. The mere impression of simplicity which a metric might give is not of course on its own sufficient; thus, for example, the relatively complicated metric (31.1) in fact has more symmetries than the ‘simple’ plane wave (29.39). Rather, we must define a symmetry in a manner independent of the coordinate system. Here we shall restrict ourselves to continuous symmetries, ignoring discrete symmetry operations (for example, space reflections).

Killing vectors

The symmetry of a system in Minkowski space or in three-dimensional (Euclidean) space is expressed through the fact that under translation along certain lines or over certain surfaces (spherical surfaces, for example, in the case of spherical symmetry) the physical variables do not change. One can carry over this intuitive idea to Riemannian spaces and ascribe a symmetry to the space if there exists an s-dimensional (1 ≤ s ≤ 4) manifold of points which are physically equivalent: under a symmetry operation, that is, a motion which takes these points into one another, the metric does not change.

What is a cosmological model?

A cosmological model is a model of our universe which, taking into account and using all known physical laws, predicts (approximately) correctly the observed properties of the universe, and in particular explains in detail the phenomena in the early universe. Such a model must also explain inter alia why the universe was so homogeneous and isotropic at the epoch of last scattering of the cosmic microwave background, and how and when inhomogeneities (galaxies and stars) arose.

In a more restricted sense cosmological models are exact solutions of the Einstein field equations for a perfect fluid that reproduce the important features of our universe. Because there is only one actual universe the large number of known or possible cosmological models may at first seem surprising. There are, however, two reasons for this multiplicity.

Firstly, only a section of our universe is known, both in space and in time. All cosmological models which differ only near the origin of the universe must be accepted for competition. In fact solutions are known which are initially inhomogeneous or anisotropic to a high degree, and which then increasingly come to approximate a Friedmann universe. All cosmological models which yield a redshift and a cosmic background radiation can hardly be refuted. The possibility cannot be excluded that our universe is not homogeneous and isotropic, but has those properties only approximately in our neighbourhood. An expanding ‘dust star’, that is, a section of a Friedmann universe which is surrounded externally by a static Schwarzschild metric (the model of a collapsing star discussed in Section 36.3), may also perhaps be an excellent model of the universe.

Some general remarks

The gravitational fields of the Earth and the Sun constitute our natural environment and it is in these fields that the laws of gravity have been investigated and summed up by equations. Both fields are to good approximation spherically symmetric and, as a result, suitable objects to test the Einstein theory as represented in the Schwarzschild metric.

The Einstein theory contains the Newtonian theory of gravitation as a first approximation and in this sense is of course also confirmed by Kepler's laws. What chiefly interests us here, however, are the – mostly very small – corrections to the predictions of the Newtonian theory. In very exact experiments one must distinguish carefully between the following sources of deviation from the Newtonian spherically symmetric field:

1. (a) Relativistic corrections to the spherically symmetric field,

2. (b) Newtonian corrections, due to deviations from spherical symmetry (flattening of the Earth or Sun, taking into account the gravitational fields of other planets),

3. (c) Relativistic corrections due to deviations from spherical symmetry and staticity.

The Newtonian corrections (b) are often larger than the relativistic effects (a) which are of interest to us here, and can be separated from them only with difficulty. Except for the influence of the rotation of the Earth (Lense–Thirring effect, see Section 27.5), one can almost always ignore the relativistic corrections of category (c).

The Michelson experiment

At the end of the nineteenth century, it was a common belief that light needs and has a medium in which it propagates: light is a wave in a medium called ether, as sound is a wave in air. This belief was shattered when Michelson (1881) tried to measure the velocity of the Earth on its way around the Sun. He used a sensitive interferometer, with one arm in the direction of the Earth's motion, and the other perpendicular to it. When rotating the instrument through an angle of 90°, a shift of the fringes of interference should take place: light propagates in the ether, and the velocity of the Earth had to be added that of the light in the direction of the respective arms. The result was zero: there was no velocity of the Earth with respect to the ether.

This negative result can be phrased differently. Since the system of the ether is an inertial system, and that of the Earth is moving with a (approximately) constant velocity, the Earth's system is an inertial system too. So the Michelson experiment (together with other experiments) tells us that the velocity of light is the same for all inertial systems which are moving with constant velocity with respect to each other (principle of the invariance of the velocity of light). The speed of light in empty space is the same for all inertial systems, independent of the motion of the light source and of the observer.

The evolutionary phases of a spherically symmetric star

In our universe a star whose temperature lies above that of its surroundings continuously loses energy, and hence mass, mainly in the form of radiation, but also in explosive outbursts of matter. Here we want to sketch roughly the evolution of such a star which is essentially characterized and determined by the star's innate properties (initial mass and density, …) and its behaviour in the critical catastrophic phases of its life.

According to observation, stars exist for a very long time after they have formed from hydrogen and dust. Therefore they can almost always settle down to a relatively stable state in the interplay between attractive gravitational force, repulsive (temperature-dependent) pressure and outgoing radiation.

The first stable state is reached when the gravitational attraction has compressed and heated the stellar matter to such a degree that the conversion of hydrogen into helium is a long-term source of energy sufficient to prevent the star cooling and to maintain the pressure (a sufficiently large thermal velocity of the stellar matter) necessary to compensate thegravitational force. The average density of such a star is of the order of magnitude 1 g cm-3. A typical example of such a star is our Sun.

When the hydrogen of the star is used up, the star can switch over to other nuclear processes (possibly only after an unstable phase associated with explosions) and produce nuclei of higher atomic number. These processes will last a shorter time and follow one another more quickly.